Recent zbMATH articles in MSC 60G40https://zbmath.org/atom/cc/60G402021-05-28T16:06:00+00:00WerkzeugPricing of margin call stock loan based on the FMLS.https://zbmath.org/1459.912072021-05-28T16:06:00+00:00"Xiang, Kaili"https://zbmath.org/authors/?q=ai:xiang.kaili"Hu, Peng"https://zbmath.org/authors/?q=ai:hu.peng"Li, Xiao"https://zbmath.org/authors/?q=ai:li.xiaoSummary: In common stock loan, lenders face the risk that their loans will not be repaid if the stock price falls below loan, which limits the issuance and circulation of stock loans. The empirical test suggests that the log-return series of stock price in the US market reject the normal distribution and admit instead a subclass of the asymmetric distribution. In this paper, we investigate the model of the margin call stock loan problem under the assumption that the return of stock follows the finite moment log-stable process (FMLS). In this case, the pricing model of the margin call stock loan can be described by a space-fractional partial differential equation with a time-varying free boundary condition. We transform the free boundary problem to a linear complementarity problem, and the fully-implicit finite difference method that we used is unconditionally stable in both the integer and fractional order. The numerical experiments are carried out to demonstrate differences of the margin call stock loan model under the FMLS and the standard normal distribution. Last, we analyze the impact of key parameters in our model on the margin call stock loan evaluation and give some reasonable explanation.Where should you park your car? The \(\frac{1}{2}\) rule.https://zbmath.org/1459.760142021-05-28T16:06:00+00:00"Krapivsky, P. L."https://zbmath.org/authors/?q=ai:krapivsky.pavel-l"Redner, S."https://zbmath.org/authors/?q=ai:redner.sidneyExistence and uniqueness of viscosity solutions for nonlinear variational inequalities associated with mixed control.https://zbmath.org/1459.490052021-05-28T16:06:00+00:00"Hu, Shipei"https://zbmath.org/authors/?q=ai:hu.shipeiSummary: The author investigates the nonlinear parabolic variational inequality derived from the mixed stochastic control problem on finite horizon. Supposing that some sufficiently smooth conditions hold, by the dynamic programming principle, the author builds the Hamilton-Jacobi-Bellman (HJB for short) variational inequality for the value function. The author also proves that the value function is the unique viscosity solution of the HJB variational inequality and gives an application to the quasi-variational inequality.\(p\)-conformal maps on the triangular lattice.https://zbmath.org/1459.600922021-05-28T16:06:00+00:00"Akahori, Jirô"https://zbmath.org/authors/?q=ai:akahori.jiro"Ida, Yuuki"https://zbmath.org/authors/?q=ai:ida.yuuki"Markowsky, Greg"https://zbmath.org/authors/?q=ai:markowsky.greg-tSummary: In [\textit{J. Akahori} et al., Stat. Probab. Lett. 151, 42--48 (2019; Zbl 07101624)], \(p\)-conformal (or \textit{Parisian}-conformal) maps on the triangular lattice were defined. The definition of \(p\)-conformality is nonstandard in comparison to ordinary discrete derivatives, but was seen to be natural in connection with a particular type of random walk, the \textit{Parisian random walk}. In this note, we establish the fact that the only \(p\)-conformal polynomials in \(z\) and \(\overline{z}\) on the triangle lattice are linear combinations of 1, \(z\) and \(z^2 - \overline{z}\), but that if one extends the notion of \(p\)-conformality to functions of two variables (the complex variable \(z\) and the time variable \(t\)) we obtain a rich class of polynomials which yield martingales when applied to the Parisian walk. These polynomials make use of a particular type of martingale transform, which is defined in the paper.Utility maximization when shorting American options.https://zbmath.org/1459.912092021-05-28T16:06:00+00:00"Zhou, Zhou"https://zbmath.org/authors/?q=ai:zhou.zhouA computational weighted finite difference method for American and barrier options in subdiffusive Black-Scholes model.https://zbmath.org/1459.912202021-05-28T16:06:00+00:00"Krzyżanowski, Grzegorz"https://zbmath.org/authors/?q=ai:krzyzanowski.grzegorz-piotr"Magdziarz, Marcin"https://zbmath.org/authors/?q=ai:magdziarz.marcinIn this paper the authors derive the linear complementarity problem (LCP) system describing the fair price of an American option in subdiffusive Black-Scholes (B-S) model.
The weighted scheme of the weighted finite difference (FD) method and the Longstaff-Schwartz method to solve the system numerically are considered. Some comparisons are also given.
The following statement is precisely proved
Theorem. The fair price an American put option in the subdiffusive (B-S) model is equal to \(\nu(z,t)\), where \(\nu(z,t)\) satisfies: \[ \begin{array}{l} x=\ln z\\
u(x,t)=\nu (e^x,T-t) \end{array} \] and \(u(x,t)\) is the solution of the system \[ \begin{array}{l} u(x,0)=\max (K-exp(x),0)\\
u(x,t)=\max (K-exp(x),0),\: \: t\in [0,T]\\
\begin{array}{l} c\\
o \end{array} D^{\alpha}u(x,t)-\frac{\sigma ^2}{2}\frac{\partial ^2 u(x,t)}{\partial x^2}-(r-\frac{\sigma ^2}{2})\frac{\partial u(x,t)}{\partial x}+ru(x,t)\ge 0,\: t\in (0,T)\\
(u(x,t)-\max (K-exp(x),0))(\begin{array}{l} c\\
o \end{array} D^{\alpha}u(x,t)-\frac{\sigma ^2}{2}\frac{\partial ^2 u(x,t)}{\partial x^2}-\\
(r-\frac{\sigma ^2}{2})\frac{\partial u(x,t)}{\partial x}+ru(x,t))=0,\: \: t\in (0,T)\\
{\displaystyle \lim_{x\to \infty}}u(x,t)=0\\
{\displaystyle \lim_{x\to -\infty}}u(x,t)=K. \end{array} \] The numerical techniques presented in this paper can successfully be repeated for other time fractional diffusion models with moving boundaries problems.
Reviewer: Nikolay Kyurkchiev (Plovdiv)Functional inequalities for forward and backward diffusions.https://zbmath.org/1459.601562021-05-28T16:06:00+00:00"Bartl, Daniel"https://zbmath.org/authors/?q=ai:bartl.daniel"Tangpi, Ludovic"https://zbmath.org/authors/?q=ai:tangpi.ludovicSummary: In this article we derive Talagrand's \(T_2\) inequality on the path space w.r.t. the maximum norm for various stochastic processes, including solutions of one-dimensional stochastic differential equations with measurable drifts, backward stochastic differential equations, and the value process of optimal stopping problems.
The proofs do not make use of the Girsanov method, but of pathwise arguments. These are used to show that all our processes of interest are Lipschitz transformations of processes which are known to satisfy desired functional inequalities.